Problem 14 - Elementary Algebra

Question

For the following problems, write the equation of the line using the given information in slope-intercept form. $$ m=3, y \text { -intercept }(0,4) $$

Step-by-Step Solution

Answer

Answer: The equation of the line is y = 3x + 4.

Step 1: Identify the given slope and y-intercept

The given slope (m) is 3, and the given y-intercept is (0, 4).

Step 2: Use the slope-intercept form

The slope-intercept formula is: $$ y = mx + b $$

Step 3: Plug in the given values

Replace m with 3, and b with 4 in the formula: $$ y = 3x + 4 $$

Step 4: Write the final equation

The equation of the line in slope-intercept form is: $$ y = 3x + 4 $$

Key Concepts

Linear Equations

Linear equations form the backbone of algebra. These equations represent straight lines on a graph and are pivotal in understanding the relationship between variables. A typical linear equation has the form \( ax + by = c \), where \( x \) and \( y \) represent variables, and \( a, b, \) and \( c \) are constants.

What makes linear equations special is their constant rate of change, meaning they have a consistent slope throughout. This makes them a key tool in predicting and understanding patterns.

• Linear equations graph as straight lines.
• They showcase a direct relationship between two variables.
• They are used in various fields to model real-world scenarios.

Understanding linear equations is like learning the rules of a game: once you know them, solving problems becomes much simpler. The solution to a linear equation like the one in slope-intercept form typically points to a specific point or a set of points on the graph.

Y-Intercept

The y-intercept is a vital concept within the context of linear equations and graphs. The y-intercept is the point where the line crosses the y-axis. It shows the value of \( y \) when \( x \) is zero. In the slope-intercept formula \( y = mx + b \), the \( b \) is the y-intercept.

To understand it better:

• The y-intercept provides a starting point to draw the line on a graph.
• It represents the output of the function at the beginning, when no input (\( x = 0 \)) has been evaluated yet.
• In our example, the y-intercept is 4, reflecting the point (0, 4) on the graph.

Recognizing the y-intercept allows you to mark a fixed point on the graph immediately, which is crucial for graphing the entire line accurately.

Slope

The slope of a line is a key component in understanding its behavior and direction. The slope indicates how steep a line is and in which direction it increases or decreases. Mathematically, the slope is represented by \( m \) in the slope-intercept form \( y = mx + b \).

The slope is calculated as the "rise over run," which is the ratio of vertical change to horizontal change between two points on the line.

• A positive slope, such as our example’s slope of 3, means the line ascends as it moves from left to right.
• A negative slope suggests the line descends from left to right.
• A slope of zero indicates a perfectly horizontal line, showing no vertical change as \( x \) changes.

Understanding slope helps predict and visualize how variables are related in various situations. It’s like knowing the incline of a hill you’re about to climb: critical for anticipating effort needed in your calculations.